DETERMINATION OF THE PARAMETERS OF CANCELLOUS BONE USING LOW FREQUENCY ACOUSTIC MEASUREMENTS

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1 Journal of Computational Acoustics, Vol. 12, No. 2 (2004) c IMACS DETERMINATION OF THE PARAMETERS OF CANCELLOUS BONE USING LOW FREQUENCY ACOUSTIC MEASUREMENTS JAMES L. BUCHANAN Mathematics Department, United States Naval Academy Annapolis MD, USA ROBERT P. GILBERT Department of Mathematical Sciences, University of Delaware, Newark DE, USA KHALDOUN KHASHANAH Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken NJ 07030, USA Received 29 May 2002 The Biot model is widely used to model poroelastic media. Several authors have studied its applicability to cancellous bone. In this article the feasibility of determining the Biot parameters of cancellous bone by acoustic interrogation using frequencies in the 5 15 khz range is studied. It is found that the porosity of the specimen can be determined with a high degree of accuracy. The degree to which other parameters can be determined accurately depends upon porosity. Keywords: Osteoporosis; cancellous bone; poroelastic media; Biot model; inverse problem, finite elements; simplex method. 1. Introduction Cancellous bone is a two component material consisting of a calcified bone matrix with interstial fatty marrow. Hence mathematical models of poroelastic media are applicable. McKelvie and Palmer, 6 Williams, 7 and Hosokawa and Otani 5 discuss the application of Biot s model for a poroelastic medium to cancellous bone. Use of this model requires determination of the parameters upon which it depends. This can be an expensive process. In this article we investigate whether these parameters can be ascertained by acoustic interrogation. 2. The Biot Model Applied to Cancellous Bone The Biot model treats a poroelastic medium as an elastic frame with interstial pore fluid. Cancellous bone is anisotropic, however, as pointed out by Williams, if the acoustic waves passing through it travel in the trabecular direction an isotropic model may be acceptable. We will simulate a two dimensional version of the experiments described in McKelvie and Palmer and Hosokawa and Otani. The motion of the frame and fluid within the bone are 99

2 100 J. L. Buchanan, R. P. Gilbert & K. Khashanah tracked by position vectors u = [u, v] and U = [U, V ]. The constitutive equations used by Biot are those of a linear elastic material with terms added to account for the interaction of the frame and interstial fluid σ xx = 2µe xx + λe + Qɛ, σ yy = 2µe yy + λe + Qɛ, σ xy = µe xy, σ yx = µe yx, σ = Qe + Rɛ where the solid and fluid dilatations are given by The stress-strain relations are e = u = u x + v y, e xx = u x, e xy = e yx = u y + v x, (1) ɛ = U = U x + V y. (2) e yy = v y. (3) The parameter µ, the complex frame shear modulus is measured. The other parameters λ, R and Q occurring in the constitutive equations are calculated from the measured or estimated values of the parameters given in Table 1 using the formulas λ = K b 2 3 µ + (K r K b ) 2 2βK r (K r K b ) + β 2 K 2 r D K b where β2 K 2 r R = D K b Q = βk r((1 β)k r K b ) D K b. (4) D = K r (1 + β(k r /K f 1)). (5) Table 1. Parameters in the Biot model. Symbol Parameter ρ f Density of the pore fluid ρ r Density of frame material K b Complex frame bulk modulus µ Complex frame shear modulus K f Fluid bulk modulus K r Frame material bulk modulus β Porosity η Viscosity of pore fluid k Permeability α Structure constant a Pore size parameter

3 Determination of the Parameters of Cancellous Bone 101 The bulk and shear moduli K b and µ are often given imaginary parts to account for frame inelasticity. Equations (1), (2) and (3) and an argument based upon Lagrangian dynamics are shown in Ref. 2 to lead to the following equations of motion for the displacements u, U and dilatations e, ɛ, µ 2 u + [(λ + µ)e + Qɛ] = 2 t 2 (ρ 11u + ρ 12 U) + b (u U) t [Qe + Rɛ] = 2 t 2 (ρ 12u + ρ 22 U) b (u U). t Here ρ 11 and ρ 22 are density parameters for the solid and fluid, ρ 12 is a density coupling parameter, and b is a dissipation parameter. These are calculated from the inputs of Table 1 using the formulas where ρ 11 = (1 β)ρ r β(ρ f mβ) ρ 12 = β(ρ f mβ) ρ 22 = mβ 2 b = F (a ωρ f /η)β 2 η k m = αρ f β and the multiplicative factor F (ζ), which was introduced in Ref. 3 to correct for the invalidity of the assumption of Poiseuille flow at high frequencies, is given by F (ζ) = 1 ζt (ζ) 4 1 2T (ζ)/iζ where T is defined in terms of Kelvin functions T (ζ) = ber (ζ) + ibei (ζ) ber(ζ) + ibei(ζ). The bone specimen is assumed to oscillate harmonically in time: u(x, y, t) = u(x, y)e iωt, U(x, y, t) = U(x, y)e iωt. Substituting these representations into (6) gives (6) (7) µ 2 u + [(λ + µ)e + Qɛ] + p 11 u + p 12 U = 0 [Qe + Rɛ] + p 12 u + p 22 U = 0 (8) where p 11 := ω 2 ρ 11 iωb, p 12 := ω 2 ρ 12 + iωb, p 22 := ω 2 ρ 22 iωb. (9) The article of McKelvie and Palmer contains estimates of the Biot parameters of cancellous bone in the human os calcis (heel bone) for the normal (β = 0.72) and severely

4 102 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 2. Estimated values of some Biot parameters at different porosities taken from McKelvie and Palmer or Hosokawa and Otani. The second set of values for the permeability were calculated from the indicated value of the pore size parameter using the Kozeny Carmen equation. β k a α Re K b Re µ , , , , osteoporotic (β = 0.95) cases while the article of Hosokawa and Otani has estimates for bovine femoral bone for porosities of β = 0.75, 0.81 and The question we shall address is whether these parameters can be recovered by acoustic interrogation of the specimen. Table 2 contains estimates of these six Biot parameters for five bone specimens. In obtaining them we have followed the estimation procedures used by McKelvie and Palmer and Hosokawa and Otani: The real parts of K b and µ were calculated using the formulas of Williams Re K b = Re µ = E 3(1 2ν) V f n E 2(1 + ν) V f n used by Hosokawa and Otani. Here V f = 1 β is the bone volume fraction. Theoretically n = 1 for waves travelling in the trabecular direction and is between 2 and 3 for transverse waves, however there is enough randomness in the trabecular direction in bone that authors have empirically adjusted the exponent to agree better with experiment. Williams arrived at a value of n = 1.23 based on comparing the Biot predictions for Type I compressional and shear wave velocity assuming the form (10) to the measured speeds obtained from experiments conducted on samples taken from bovine tibia. Hosokawa and Otani found that n = 1.46 agreed well with their data from experiments on bone specimens from bovine femora. We shall use the exponent of Hosokawa and Otani and also their values E = , ν = 0.32 for the Young s modulus and Poisson ratio of solid bone. The imaginary parts of K b and µ were calculated using a log decrement l : Im K b = l Re K b /π, Im µ = l Re µ /π with a value l = 0.1 which is typical of that used in underwater acoustics. There appears to be little sensitivity to these parameters, however. The structure factor was calculated using the formula of Berryman α = 1 r(1 1/β) with r = 0.25, again following Williams and Hosokawa and Otani. The pore size parameter was estimated by McKelvie and Palmer using electron microscopy and by Hosokawa and Otani using x-ray examination. Figure 1 shows that their estimates indicate that pore size is approximately a linear function of porosity. (10)

5 Determination of the Parameters of Cancellous Bone x Pore size Porosity Fig. 1. Estimated values of the pore size parameter ( ) for five bone specimens along with the regression line. Permeability is a difficult parameter to estimate. Figure 2 shows the values for the five specimens of McKelvie and Palmer and Hosokawa and Otani. The estimates indicate that permeability is approximately a log-linear function of porosity. However McKelvie and Palmer characterize their values without elaboration as estimates and Hosokawa and Otani state that their estimates are based upon those of McKelvie and Palmer. Hence the apparent log-linear relation should be regarded circumspectly. Indeed according to the Kozeny Carmen formula k = βa2 4K, (11) where K 5 is an empirical constant, the relation is not log-linear. Figure 2 shows that if pore size is indeed a linear function of porosity as indicated by Fig. 1, then permeability, as predicted by (11), will deviate significantly from log-linearity. Table 3 gives the values we shall use for the other Biot parameters. The value for ρ f is from McKelvie and Palmer, but the value used by Hosokawa and Otani, 930, is similar. Likewise both sets of authors used about the same value for viscosity η. The fluid bulk modulus K f is from Hosokawa and Otani. The frame material densities used by McKelvie and Palmer and Hosokawa and Otani were somewhat different. Williams reports a range of estimates for bovine cortical bone of ρ r = We follow Williams and Hosokawa and Otani in using ρ r = The frame material bulk modulus was calculated from (10) with V f = 1.

6 104 J. L. Buchanan, R. P. Gilbert & K. Khashanah Permeability Porosity Fig. 2. Estimated values of permeability ( ) for five bone specimens. Dashed line: regression line. Solid line: Value of premeability predicted by the Kozeny Carmen equation assuming a linear relation between pore size and porosity. Table 3. Parameters for cancellous bone to be used for all specimens. Parameter Symbol Value Pore fluid density ρ f 950 Fluid bulk modulus K f Pore fluid viscosity η 1.5 Frame material density ρ r 1960 Frame material bulk modulus K r The question we shall address is whether it is feasible to recover some of the Biot parameters by measuring the acoustic field arising from a point source placed in a tank of water containing a specimen of bone. Based on the discussion above the parameters we shall seek to recover are the ones concerning which there is the most uncertainty: porosity β, permeability k, pore size a, structure factor α and the real parts of the bulk and shear frame moduli K b and µ. 3. Finite Element Formulation of the Problem A bone specimen is placed in a water tank. The region occupied by the bone specimen and the water are Ω b and Ω w respectively. In Ω w we have in the two-dimensional case the

7 Determination of the Parameters of Cancellous Bone 105 differential equations for fluid pressure P and displacement [U w, V w ] 2 P k 2 0 P = S(x, y, x 0, y 0 ) P + ρ w ω 2 [U w, V w ] = 0, (12) assuming a source S located at (x 0, y 0 ). Multiplying by a test function and applying the divergence theorem gives ( P ψ k0p 2 ψ)da n w P ψds = s(x, y, x 0, y 0 )ψda Ω w Ω w Ω w where the unit normal vector n w points into the bone. In two dimensions the Eq. (8) are (λ + 2µ) xx u + µ yy u + (λ + µ) xy v + Q xx U + Q xy V + p 11 u + p 12 U = 0 (λ + 2µ) yy v + µ xx v + (λ + µ) xy u + Q yy V + Q xy U + p 11 v + p 12 V = 0 Q x ( x u + y v) + R x ( x U + y V ) + p 12 u + p 22 U = 0 Q y ( x u + y v) + R y ( x U + y V ) + p 12 v + p 22 V = 0 (13) The finite elements package FEMLAB was used for the computations in this article. In FEMLAB systems of partial differential equations are written xj (c lkji xi u k + α lkj u k + γ lj ) + β lki xi u k + a lk u k = f l (14) with the summation notation convention in effect. For the Biot equations without a source α, β, γ, f = 0 which gives xj (c lkji xi u k ) + a lk u k = 0. (15) Multiplying (15) by a test function φ and integrating over Ω b gives Ω b ( xj (c lkji xi u k ) + a lk u k )φda = 0. Applying the divergence theorem gives (c lkji xi u k xj φ + a lk u k φ)da n bj (c lkji xi u k )φds = 0, l = 1, 2, 3, 4 Ω b Ω b where n b = (n bj ) is the outward unit normal from Ω b. We consider the two dimensional case x 1 = x, x 2 = y and take u 1 = u, u 2 = v, u 3 = U, u 4 = V. The stress tensor T jl = c lkji xi u k must be chosen appropriately for the interface conditions. At the water-bone interface the following conditions are required (cf. Ref. 4) continuity of flux: [U w, V w ] = β[u, V ] + (1 β)[u, v] continuity of stress: P = σ xx + σ, P = σ yy + σ, σ xy = σ yx = 0 continuity of pore fluid pressure: βp = σ.

8 106 J. L. Buchanan, R. P. Gilbert & K. Khashanah This suggests we take [ ] σxx + σ σ xy σ 0 T =. σ yx σ yy + σ 0 σ It then follows from (1), (2) and (3) that xj T j1 = x (σ xx + σ) + y σ yx = x (2µ x u + λ( x u + y v) + Q( x U + y V ) + Q( x u + y v) + R( x U + y V )) + y µ( y u + x v) = 2µ xx u + λ xx u + λ xy v + Q xx U + Q xy V + Q xx u + Q xy v + R xx U + R xy V + µ yy u + µ xy v = (λ + 2µ + Q) xx u + (λ + Q + µ) xy v + (R + Q) xx U + (R + Q) xy V + µ yy u xj T j3 = x σ = Q( xx u + xy v) + R( xx U + xy V ) and similarly for xj T j2 and xj T j4. Adding (13) 3 to (13) 1 and (13) 4 to (13) 2 gives the desired form of the equations Thus we want (λ + 2µ + Q) xx u + µ yy u + (λ + µ + Q) xy v + (R + Q) xx U + (R + Q) xy V + (p 11 + p 12) u + (p 12 + p 22 )U = 0 (λ + 2µ + Q) yy v + µ xx v + (λ + µ + Q) xy u + (R + Q) yy V + (R + Q) xy U + (p 11 + p 12 )v + (p 12 + p 22 )V = 0 Q x ( x u + y v) + R x ( x U + y V ) + p 12 u + p 22 U = 0 Q y ( x u + y v) + R y ( x U + y V ) + p 12 v + p 22 V = 0. T 11 = c 1k1i xi u k = c 1111 x u + c 1112 y u + c 1211 x v + c 1212 y v + c 1311 x U + c 1312 y U + c 1411 x V + c 1412 y V = σ xx + σ = (λ + 2µ + Q) x u + (λ + Q) y v + (Q + R) x U + (Q + R) y V c 1111 = λ + 2µ + Q, c 1212 = λ + Q, c 1311 = c 1412 = Q + R T 21 = c 1k2i xi u k = c 1121 x u + c 1122 y u + c 1221 x v + c 1222 y v + c 1321 x U + c 1322 y U + c 1421 x V + c 1422 y V = σ yx = µ( y u + x v) c 1122 = c 1221 = µ

9 Determination of the Parameters of Cancellous Bone 107 T 12 = c 2k1i xi u k = c 2111 x u + c 2112 y u + c 2211 x v + c 2212 y v + c 2311 x U + c 2312 y U + c 2411 x V + c 2412 y V = σ xy = µ( x v + y u) c 2112 = c 2211 = µ T 22 = c 2k2i xi u k = c 2121 x u + c 2122 y u + c 2221 x v + c 2222 y v + c 2321 x U + c 2322 y U + c 2421 x V + c 2422 y V = σ yy + σ = (λ + 2µ + Q) y v + (λ + Q) x u + (Q + R) x U + (Q + R) y V c 2121 = λ + Q, c 2222 = λ + 2µ + Q, c 2321 = c 2422 = Q + R T 13 = c 3k1i xi u k = c 3111 x u + c 3112 y u + c 3211 x v + c 3212 y v + c 3311 x U + c 3312 y U + c 3411 x V + c 3412 y V = σ = Q( x u + y v) + R( x U + y V ) c 3111 = c 3212 = Q, c 3311 = c 3412 = R T 24 = c 4k2i xi u k = c 4121 x u + c 4122 y u + c 4221 x v + c 4222 y v + c 4321 x U + c 4322 y U + c 4421 x V + c 4422 y V = σ = Q( x u + y v) + R( x U + y V ) c 4121 = c 4222 = Q, c 4321 = c 4422 = R This gives ( ) ( ) λ + 2µ + Q 0 0 λ + Q c 11.. =, c 12.. = 0 µ µ 0 ( ) ( ) Q + R 0 0 Q + R c 13.. =, c 14.. = ( ) ( ) 0 µ µ 0 c 21.. =, c 22.. = λ + Q 0 0 λ + 2µ + Q ( ) ( ) c 23.. =, c 24.. = Q + R 0 0 Q + R ( ) ( ) Q 0 0 Q c 31.. =, c 32.. =

10 108 J. L. Buchanan, R. P. Gilbert & K. Khashanah Also ( ) ( ) R 0 0 R c 33.. =, c 34.. = ( ) ( ) c 41.. =, c 42.. = Q 0 0 Q ( ) ( ) c 43.. =, c 44.. =. R 0 0 R a 11 = a 22 = (p 11 + p 12 ), a 13 = a 24 = (p 12 + p 22 ) a 31 = a 42 = p 12, a 33 = a 44 = p 22. In FEMLAB interface conditions are of the form n (c u) + q u = 0 where for this problem u = [P, u, v, U, V ] T. The interface conditions are Continuity of flux: From (12) 2 and thus n w P = n w ρ w ω 2 [U w, V w ] = n w ρ w ω 2 (β[u, V ] + (1 β)[u, v]). Here n w points into the bone. Continuity of stress: n w P + n w ρ w ω 2 (β[u, V ] + (1 β)[u, v]) = 0. n bj T jl + n bl P = 0, l = 1, 2 since an expansion of the bone induces a compression (P < 0) in the water. Here n b points into the water. Continuity of pore pressure: n bj T jl n bl βp = 0, l = 3, 4. This gives 0 ρ w ω 2 (1 β)n wx ρ w ω 2 (1 β)n wy ρ w ω 2 βn wx ρ w ω 2 βn wy n bx P q = n by P n bx βp n by βp

11 Determination of the Parameters of Cancellous Bone Preliminary Explorations We want to simulate the following experiment: a rectangular bone specimen of the approximate dimensions of the ones used in the experiment described in Hosokawa and Otani, m is placed in a tank of water. A time-harmonic point source is located at the origin. The tank is open at the top whence we use a pressure release condition P = 0. The sides of the tank are assumed to be perfectly reflecting P/ n = 0. The dimensions of the tank will be based upon a distance s: The bone specimen is centered on the x-axis a distance 2s to the right of the origin. The left edge of the tank is at x = s. The surface of the water is a distance 2s above the top of the bone, the bottom of the tank a distance 2s below the bottom of the bone and the right edge of the tank a distance 2s from the right edge of the bone. We will experiment with different values of s. We shall assume that the acoustic field generated by the point source has values Pij at points (x i, y j ), i = 1,..., L, j = 1,..., M located in the tank. A set of Biot parameters which produces trial values P ij will be compared to the measured values using the objective function f(p ij, P ij) = L i=1 P i. P i. 2 L i=1 Pi. 2. (16) Thus the problem is minimize the objective function. The minimization will be carried out using the Nelder Mead simplex algorithm. The process is complicated by the various errors involved: (1) in an actual experiment there would be errors in the measurements Pij, (2) the trial values P ij are calculated from the Biot model and thus will be in poor agreement with the measured values to the extent that the Biot model is a poor approximation to cancellous bone, and (3) since the trials values will be calculated using the finite element method, there will be discretization error as well. Lacking experimental data, we are not in a position to assess the effects of the first two types of error. Consequently we will focus on the third type. To assess it we will calculate the measured data Pij assuming the Biot model and using a finite elements mesh which is the third refinement of FEMLAB s initial mesh, but use only two refinements when calculating the trial fields P ij. Figure 3 shows the result of calculating the objective function f(p ij, Pij ) for the size parameters s = 0.03, 0.07 and s = 0.10 m with measurements taken at 10 points in the middle 80% of the tank along two vertical lines, one midway between the source and the bone specimen, the other midway between the bone and the right edge of the tank. Both P ij and Pij were calculated using the Biot parameters given for porosity β = 0.72 in Tables 2 and 3, but P ij was calculated using two refinements of the initial finite element mesh whereas Pij was calculated with three refinements. As can be seen the agreement between the two fields varies considerably with frequency, suggesting that the expediency and the success in recovering the Biot parameters of the bone specimen may depend on the interrogating frequency. The values of the objective function are large above about 7 khz for s = 0.10 and

12 110 J. L. Buchanan, R. P. Gilbert & K. Khashanah s = 0.10 s = 0.07 s = Frequency (Hz) Fig. 3. Objective function values for three tank sizes. The target pressure data was calculated using the finite element method with three refinements, the trial data with two refinements. The porostiy of the specimen was about 10 khz for s = 0.07 suggesting that calculating the trial data with two refinements of the initial mesh may be inadequate for tanks with dimensions on the order of a half meter on a side. While the agreement is good at frequencies in the low kilohertz range the question arises as to whether this is simply due to little interaction with the bone specimen. To investigate the influence of the interrogating frequency and the effect of using two refinements of the finite elements mesh for the trial data, but three refinements for the simulated data, we attempted to recover six Biot parameters starting with guesses that were in fact the correct values for the target specimen. Table 4 shows the results of attempting to recover the six parameters β, k, a, α, Re K b and Re µ using the Nelder Mead simplex method to perform a multivariate minimization on the objective function (16) at different frequencies when the tank size parameter was s = As can be seen the real part of the bulk modulus became negative at 1 khz, suggesting that interaction with the bone at low frequencies may be insufficient to determine some parameters. The three frequencies, and were chosen because they represent respectively the cases where for β = 0.72 the agreement in pressure between two and three refinements of the finite element mesh was good, intermediate and poor (cf. Fig. 3). For these three frequencies between 5 and 10 khz there was no easily discernible pattern as to which frequency produced the best results for a particular parameter. It may be noted however that the porosity estimates were somewhat more accurate at Hz where the agreement for two and three refinements of the finite element mesh was best. Also noteworthy is that the pore size parameter was in three instances off by a factor of more than two, indicating this parameter may not be

13 Determination of the Parameters of Cancellous Bone 111 Table 4. Results of an application of the simplex method when the initial guess for the parameters was the correct one for the target specimen. Three refinements of the initial finite element mesh were used for the target data, two for the trial data. Frequency f(p ij, P ij ) β k a α Re K b Re µ Guess/Target Guess/Target Guess/Target Guess/Target Table 5. Number of iterations and evaluations of the objective function and final value of the objective function resulting from an application of the simplex method, the results of which are shown in Table 4. Frequency β Iterations Evaluations f(p ij, P ij ) f min

14 112 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 6. Results of an application of the simplex method when the initial guesses for the parameters were those for cancellous bone of porosity 0.81 (see Table 2). Three refinements of the initial finite element mesh were used for the target data, two for the trail data. Frequency β k a α Re K b Re µ Guess Target Target Target Target strongly influential and that, except for bone of porosity 0.95 the structure factor α drifted far from its initial (correct) value indicating that this parameter is not influential for bone at the lower porosities. Table 5 indicates that there was wide variance in the number of iterations and evaluations of the objective function the simplex method required to converge. There was no strong correlation between frequency and efficiency for the three frequencies tested. It may be noted however that at Hz where the initial agreement was best, the reduction of the value of the objective function was least. Since reduction of the objective function below the value produced by the correct values of the parameter when two refinements of the finite element mesh are used represents uncertain progress, this may be a virtue. Another preliminary test that we conducted was to see if a simple scheme for supplying initial guesses for the six parameters to the simplex minimization method would suffice. Table 6 shows the results obtained when the initial guesses were the parameters for cancellous bone of porosity 0.81 and the targets were bone of porosities 0.72, 0.75, 0.83 and This simple approach worked well for the three lower porosities, but poorly for β = Hence if parameter recovery is to be successful over the entire range of porosities that are expected, a more sophisticated algorithm is required.

15 Determination of the Parameters of Cancellous Bone An Algorithm for Recovering the Biot Parameters As indicated in the last section the recovery of the Biot parameters of a bone specimen by a minimization may be successful if sufficiently good initial guesses for the parameters can be found. We will generate these guesses by first formulating the problem as a univariate minimization problem for the single parameter porosity. This is feasible because of the various formulas relating other Biot parameters to porosity discussed in Sec. 2. Since these formulas were used in creating the target Biot parameters that we are trying to recover, variations based on the uncertainties mentioned were incorporated where possible. For a given value of porosity β Re K b (β) and Re µ(β) are determined from the formulas (10). Since the values for the target specimens of Table 2 were determined using Hosokawa and Otani s value n = 1.46 we used Williams value n = 1.23 in calculating the trail data in the univariate minimization process. Pore size a(β) is determined by the regression line in Fig. 1. Since this parameter was measured independently by McKelvie and Palmer and Hosokawa and Otani for their bone specimens and the results indicate an approximately linear relation between pore size and porosity, this is justifiable. Permeability k(β) is determined from the regression line shown in Fig. 2. Since, as indicated earlier, there is some uncertainty about the log-linear relationship indicated by Fig. 1, we shall also test the algorithm when the permeability of the target specimen is calculated from the Kozeny Carmen equation (11) using the pore size parameter in Porosity 0.85 f(p,p*) Frequency (Hz) Fig. 4. Estimated porosity (solid line) and objective function value (dashed line) for bone of porosity 0.72.

16 114 J. L. Buchanan, R. P. Gilbert & K. Khashanah Porosity f(p,p*) Frequency (Hz) Fig. 5. Estimated porosity (solid line) and objective function value (dashed line) for bone of porosity Frequency (Hz) x x 10 4 Fig. 6. Solid line: values of f(p ij, Pij ) when P ij and Pij are calculated for bone specimens of porosity 0.72 (top) and 0.83 (bottom) using finite elements with two and three mesh refinements respectively. Dashed line: values of f(p ij (β min ), Pij ).

17 Determination of the Parameters of Cancellous Bone 115 Table 2. As indicated in Fig. 2 this gives only modest agreement with the estimates of McKelvie and Palmer and Hosokawa and Otani. The structure factor α(β) is determined from the formula of Berryman. 1 The value of the parameter r used was suggested by the finite element analysis of Yavari and Bedford, 8 rather than the value 0.25 used by Hosokawa and Otani and Williams, but this makes little difference in the predicted value. The values of all other Biot parameters are taken from Table 3. For any given porosity β a set of Biot parameters can be constructed as described above and from this a pressure field P ij (β) calculated using finite elements with two refinements of the initial mesh. At a given frequency the measured data Pij can be calculated using three mesh refinements and the value of β which minimizes the objective function f(p ij (β), Pij ) found. Figures 4 and 5 show the results of this univariate minimization procedure over the frequency range 1 15 khz when the target specimens had porosities 0.72 and 0.83 respectively. The minimum was sought in the range 0.60 β 0.99 and the tank size Table 7. Results of two applications of the simplex method. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β k a α Re K b Re µ f min 6000 Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result

18 116 J. L. Buchanan, R. P. Gilbert & K. Khashanah parameter was s = Except at a few frequencies the estimated porosities were close to the target values. Also shown are minimum values of the objective function at each frequency. As Fig. 6 shows these minimum values follow the trend of those for s = 0.03 in Fig. 3, and thus we have a means of finding frequencies at which the agreement between trial and measured pressure fields may be good without knowing the Biot parameters for the target specimen. Tables 7 14 show the results of using the simplex method to perform a multivariate minimization when the pressure data and Biot parameters for the target specimens were calculated different ways. The algorithm used was as follows: The univariate minimization procedure described above was used to generate initial estimates, labeled Guess 1 in the Tables, for the six Biot parameters for which values were sought. The frequency used was the one that produced the best agreement between trial and measured data in the frequency range khz (cf. Figs. 4 and 5). The Nelder Mead simplex method was initialized with these values. As can be seen in Tables 7 and 9 the initial estimates for porosity and hence the other parameters were good when the target permeabilities were the first set in Table 8. Wave speeds and attenuations resulting from two applications of the simplex method. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β c p1 c p2 c s γ p1 γ p2 γ s 6000 Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result

19 Determination of the Parameters of Cancellous Bone 117 Table 9. Results of two applications of the simplex method. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β k a α Re K b Re µ f min Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Table 2, but sometimes were poor when the second (Kozeny Carmen) set of permeabilities was used. The simplex method was always fairly successful in determining the porosity, but was sometimes less successful in determining the other parameters. The result of the first application are referred to as Result 1 in the Tables. To see whether a better initial guess for the parameters would improve the accuracy of the other parameters, a new initial guess, labeled Guess 2, for the simplex method was constructed using the porosity, bulk and shear moduli determined by the first application, but with the pore size and permeability calculated from the value of porosity found by the first application using the regression lines shown in Figs. 1 and 2. The result of this second application is referred to as Result 2 in the Tables. Tables 8 and 10 express the results in terms of an alternative set of parameters, the speeds c p1 and c p2 of type I and II compressional waves, the speed c s of shear waves, and the attenuations γ p1, γ p2 and γ s, measured in decibels per wavelength, of the three types of waves. See Ref. 4 for the details of how these quantities are computed. Tables give the results of applying the guess and refine procedure at frequencies 8 and 9 khz (cf. Table 5).

20 118 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 10. Wave speeds and attenuations resulting from two applications of the simplex method. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β c p1 c p2 c s γ p1 γ p2 γ s Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Conclusions In discussing the results of our simulations we consider the specimens of porosities 0.72, 0.75 and 0.83 separately from that of porosity 0.95 since different parameters were influential in the latter case. Tables 15 and 16 give the percentage relative errors made by the algorithm described in the preceding section in determining the five Biot parameters porosity, permeability, pore size, and real parts of the bulk and shear moduli. The structure factor is not included since was rarely determined with much accuracy and often the value found was well below its theoretical minimum of The algorithm was uniformly successful in finding the porosity to within 3%. Percentage errors for all of the remaining parameters were often higher, but the target values of these parameters varied over at least one order of magnitude. In the case when the permeability of the specimen was the first one given in Table 2 the second application of the simplex method did not improve the results, indeed as indicated by the averages in the last rows of Table 15, they were slightly worse. This is not surprising since the regression lines used in estimating the permeability and pore size

21 Determination of the Parameters of Cancellous Bone 119 Table 11. Results of two applications of the simplex method at 8 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β k a α Re K b Re µ f min Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result for Guess 1 yielded good estimates, assuming the estimate for porosity was accurate. When the target specimen had the second (Kozeny Carmen) value for permeability the situation was different. The second application of the simplex method substantially improved poor results for permeability and pore size for the specimens of porosity 0.75 and At the other two frequencies tested, 8 and 9 khz, the determination of porosity was on the whole slightly less accurate (Tables 17 20). Also the procedure failed to find a reasonable value for permeability for the specimen of porosity 0.83 at 8 khz (Table 17). The percentage errors made by the algorithm in determining the parameters of the specimen of porosity 0.95 are given in Table 21. The algorithm was successful in determining the porosity and the estimate for the structure factor was much more accurate than for the lower porosity specimens. Determination of the pore size parameter and the real parts of the bulk and shear moduli was less accurate than at the lower porosities. Determination of permeability was very inaccurate when the Kozeny Carmen values were used and therefore the initial guesses given to the simplex method were poor. Thus only the porosity seemed to be reliably ascertainable.

22 120 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 12. Results of two applications of the simplex method at 8 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β k a α Re K b Re µ f min Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result References 1. J. G. Berryman, Confirmation of biot s theory, Appl. Phys. Lett. 37 (1980) M. A. Biot, Theory of propogation of elastic waves in a fluid-saturated porous solid. I. Lower frequency range, J. Acoust. Soc. Am. 28 (1956) M. A. Biot, Theory of propogation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range, J. Acoust. Soc. Am. 28 (1956) J. L. Buchanan and R. P. Gilbert, Transmission loss in a shallow ocean over a two-layer seabed, Int. J. Solids Structures 35(34 35) (1998) A. Hosokawa and T. Otani, Ultrasonic wave propagation in bovine cancellous bone, J. Acoust. Soc. Am. 101 (1997) M. L. McKelvie and S. B. Palmer, The interaction of ultrasound with cancellous bone, Phys. Med. Biol. 10 (1991) J. L. Williams, Prediction of some experimental results by biot s theory, J. Acoust. Soc. Am. 91 (1992) B. Yavari and A. Bedford, Comparison of numerical calculations of two biot coefficients with analytical solutions, J. Acoust. Soc. Am. 90 (1991)

23 Determination of the Parameters of Cancellous Bone 121 Table 13. Results of two applications of the simplex method at 9 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β k a α Re K b Re µ f min Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result

24 122 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 14. Results of two applications of the simplex method at 9 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β k a α Re K b Re µ f min Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result Target Guess Result Guess Result

25 Determination of the Parameters of Cancellous Bone 123 Table 15. Percentage errors in determining five Biot parameters. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β k a Re K b Re µ 6000 Target Result 1 2.1% 6.2% 23.8% 0.3% 13.8% Result 2 1.9% 2.0% 19.5% 6.6% 6.2% Target Result 1 2.3% 17.4% 8.8% 3.4% 4.5% Result 2 2.3% 44.3% 67.5% 3.4% 4.5% Target Result 1 2.2% 56.3% 58.5% 22.9% 21.7% Result 2 2.8% 42.3% 17.0% 31.4% 30.5% Average Error Result 1 2.2% 26.7% 30.3% 8.9% 13.4% Result 2 2.3% 29.5% 34.7% 13.8% 13.7% Table 16. Percentage errors in determining five Biot parameters. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β k a Re K b Re µ Target Result 1 0.1% 11.1% 47.6% 13.5% 23.1% Result 2 0.1% 11.1% 47.8% 13.8% 23.1% Target Result 1 2.5% 47.9% 60.0% 43.4% 32.3% Result 2 2.5% 10.0% 2.4% 46.4% 24.6% Target Result 1 1.9% 131.5% 152.6% 39.2% 38.1% Result 2 1.4% 16.9% 32.4% 48.0% 37.2% Average Error Result 1 1.5% 63.5% 86.7% 31.2% 19.6% Result 2 1.4% 12.7% 27.5% 36.1% 28.3%

26 124 J. L. Buchanan, R. P. Gilbert & K. Khashanah Table 17. Percentage errors in determining five Biot parameters at 8 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the first set in Table 2. Frequency β k a Re K b Re µ Target Result 1 5.3% 1.0% 36.7% 6.0% 29.2% Result 2 4.9% 3.4% 22.2% 29.2% 1.5% Target Result 1 4.5% 9.4% 21.3% 8.6% 14.2% Result 2 6.3% 52.9% 60.0% 13.9% 64.5% Target Result % 686.7% 43.3% 72.1% 70.3% Result 2 2.8% 763.3% % 8.5% 8.5% Average Error Result 1 8.5% 232.4% 33.7% 28.6% 37.9% Result 2 4.6% 273.2% 364.4% 17.2% 24.8% Table 18. Percentage errors in determining five Biot parameters at 8 khz. Three refinements of the initial finite element mesh were used for the simulated data, two for the trial data. The values for permeability were the second set in Table 2. Frequency β k a Re K b Re µ Target Result 1 3.8% 2.4% 11.3% 8.2% 13.8% Result 2 3.3% 0.8% 45.6% 18.6% 16.9% Target Result 1 1.7% 6.3% 31.3% 10.5% 16.4% Result 2 2.1% 12.9% 19.9% 7.5% 9.1% Target Result 1 3.9% 157.9% 165.2% 0.7% 44.0% Result 2 3.4% 2.0% 14.1% 2.0% 35.9% Average Error Result 1 3.1% 55.5% 69.2% 6.4% 24.7% Result 2 2.9% 5.2% 26.5% 9.3% 20.6%